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Make the Cake: Preheat oven to 350 degrees F. Grease an 8-inch square baking pan.
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In a small bowl, whisk together the flour, baking soda, and salt; set aside.
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Beat the butter and sugar with an electric mixer on medium-high speed until pale and fluffy, about 3 minutes. Add the eggs, one at a time, and beat until combined. Add the bananas and vanilla extract and beat until incorporated.
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Reduce the mixer speed to low and add the flour mixture in three additions, alternating with the milk in two additions, scraping down the bowl as needed. Using a rubber spatula, stir in the chocolate chips.
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Transfer the batter to the prepared pan and smooth the top into an even layer. Bake until a toothpick inserted into the center comes out clean, 45 to 50 minutes, rotating the pan halfway through baking. Allow the cake to cool completely.
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Make the Frosting: Using an electric mixer on medium speed, beat the butter and cream cheese on medium speed until smooth, 1 to 2 minutes. Reduce the mixer speed to low and add the powdered sugar; beat until thoroughly combined, then add the vanilla extract, increase the mixer speed to medium and beat until completely smooth, an additional 1 to 2 minutes.
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Spread the frosting evenly over the top of the cake and serve. The cake can be stored, covered, at room temperature for up to 2 days, or in the refrigerator for up to 5 days.